INSITTUTO DE ESTUDIOS SUPERIORES DE TAMAPULIPAS 

RED DE UNIVERSIDADES AN?HUAC 

M?XICO MMVI 

The Population of Mexican United States. Part I  

Maple 10 Document 

Prepared by  

Prof. David Macias Ferrer 

E-mail: david.macias@iest.edu.mx 

Madero City, Mexico 

URL: http://www.geocities.com/dmacias_iest/MyPage.html 

Goals 

Given the growth of population from Mexican United States (M?xico) since 1921 to 1995: 

• To find the curve of better fits for this data through Statistics Package

 

• To find mathematical models for the population behavior using Ordinary Differential Equations and DEtools Pakage

 

• To predict the population in the years 2000 and 2005

 

Methodology 

The growth of population can be analized by curve fitting tools or differential equations. In this document we will use both methods. 

Applied Problem 

According with the National Institute of Satistics, Geography  and Informatic (INEGI) of the Mexican government, the population growth since 1921 to 1995 is given by: 

 

 

	t	Population (Millions)
		91.2

 

 

 

The graphic is: 

 

 

To find the curve of better fits for this data through Maple 10 Packages and to find mathematical models for the population behavior using Ordinary Differential Equations 

 

Curve Fitting using Statistics Package 

Using Fit and ExponentialFit commands of the Statistics Pakage, we will find a mathematical model that approaches the behavior of the population's growth. 

 

 

The vectors of the points dispersion are: 

 

 

 

 

Polynomials Fit 

Polynomials Fit of Fifth Degree: 

 

Using Fit command with a Polynomial of fifth degree, we have: 

 

(4.1.1)

 

The Polynomial Model is: 

 

(4.1.2)

 

 

 

Predictions: 

The real value of population for 2000 is 97.5 and the predicted value of the Polynomial Model for the same year is: 

 

(4.1.3)

 

The real value of population for 2005 is 103.3 and the predicted value of the Polynomial Model for the same year is: 

 

(4.1.4)

 

Exponential Fit 

Exponential Fit 

 

Using the ExponentialFit command we have: 

 

(4.2.1)

 

The Exponential Model is: 

 

(4.2.2)

 

 

 

Predictions: 

The real value of population for 2000 is 97.5 and the predicted value of the Exponential Model for the same year is: 

 

(4.2.3)

 

The real value of population for 2005 is 103.3 and the predicted value of the Exponential Model for the same year is: 

 

(4.2.4)

 

 

 

Mathematical Models with Ordinary Differential Equation 

Malthusian Model or Exponential Model 

In 1798 the Englishman Thomas Robert Malthus (1766-1834) posited a mathematical model of population growth. He model, though simple, has become a basis for most future modeling of biological populations. His essay, "An Essay on the Principle of Population," contains an excellent discussion of the caveats of mathematical modeling. 

 

When the environment is unlimited, the population growth will follow a geometric progression, doubling its size each generation. This growth can be described by a differential equation as:  

 

where is the rate of change of numbers over time, is the natural rate of increase and p(t) is the population at any instant. In this model, the parameter include the constant birth and death rates. We will use the Maple commands to find the Malthusian Model for Mexican population growth. 

 

 

 

 

(5.1.1)

 

 

(5.1.2)

 

 

(5.1.3)

 

 

(5.1.4)

 

 

(5.1.5)

 

 

(5.1.6)

 

The Malthusian Model is: 

 

(5.1.7)

 

 

(5.1.8)

 

 

 

 

 

 

Predictions: 

 

(5.1.9)

 

The real value of population for 2000 is 97.5 and the predicted value of the Malthusian Model for the same year is: 

 

(5.1.10)

 

The real value of population for 2005 is 103.3 and the predicted value of the Malthusian Model for the same year is: 

 

(5.1.11)

 

Verhulst-Pearl Model or Logistic Model 

As the exponential growth cannot be maintained indefinitely because ressources become limiting we need to add a second element limiting the rate of growth. The differential equation could be modified to take in account the dampening effect of intraspecific competition as  

 

 

where is the population size at any instant, is the intrinsic rate of increase and, is the carrying capacity, that measures the population size that can be supported by the environment at balance.This equation has been developped independently by the mathematician and biologist Belgian Pierre Francois Verhulst (1804-1849) in his essay "Notice sur la loi que la Population suit dans son Accroissement" (1938) and the geneticist and enviromentalist Raymond Pearl (1879-1940) in his work "The Biology of Population Growth" (1925). In this case, we will take and  

 

 

 

 

(5.2.1)

 

 

(5.2.2)

 

 

(5.2.3)

 

 

(5.2.4)

 

 

(5.2.5)

 

 

 

 

 

 

Predictions: 

 

(5.2.6)

 

The real value of population for 2000 is 97.5 and the predicted value of the Verhulst-Pearl Model for the same year is: 

 

(5.2.7)

 

The real value of population for 2005 is 103.3 and the predicted value of the Verhulst-Pearl Model for the same year is: 

 

(5.2.8)

 

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